On the Smoluchowski-Kramers approximation for a ... - UMD MATH

Probab. Theory Relat. Fields 135, 363–394 (2006) Digital Object Identifier (DOI) 10.1007/s00440-005-0465-0

Sandra Cerrai · Mark Freidlin

On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom Received: 21 October 2004 / Revised version: 6 May 2005 / c Springer-Verlag 2005 Published online: 12 September 2005 –
Abstract. According to the Smolukowski-Kramers approximation, we show that the solution of the semi-linear stochastic damped wave equations µutt (t, x) =
u(t, x) − ut (t, x) + b(x, u(t, x)) + QW˙ (t), u(0) = u0 , ut (0) = v0 , endowed with Dirichlet boundary conditions, converges as µ goes to zero to the solution of the semi-linear stochastic heat equation ut (t, x) =
u(t, x) + b(x, u(t, x)) + QW˙ (t), u(0) = u0 , endowed with Dirichlet boundary conditions. Moreover we consider relations between asymptotics for the heat and for the wave equation. More precisely we show that in the gradient case the invariant measure of the heat equation coincides with the stationary distributions of the wave equation, for any µ > 0.

1. Introduction The motion of a particle of a mass µ in the field b(q) + σ (q)W˙ with the damping proportional to the speed (we put the coefficient equal to 1) is described, according to the Newton law, by the equation µ µ µ µq¨t = b(qt ) + σ (qt )W˙ t − q˙t ,

µ

µ

q0 = q ∈ R n , q˙0 = p ∈ R n .

(1.1)

Here b(q) is the deterministic component of the force and σ (q)W˙ t , where W˙ t is the standard Gaussian white noise in R n and σ (q) is an n × n-matrix, is the µ stochastic part. It is well known that, for 0 < µ δ} = 0,

µ↓0

0≤t≤T

(1.3)

S. Cerrai: Dip. di Matematica per le Decisioni, Universit`a di Firenze, Via C. Lombroso 6/17, I-50134 Firenze, Italy M. Freidlin: Department of Mathematics, University of Maryland, College Park, Maryland, USA Key words or phrases: Smolukowski-Kramers approximation – Stochastic semi-linear damped wave equations – Stochastic semi-linear heat equations – Stationary distributions – Gradient systems – Invariant measures.

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for any 0 ≤ T < ∞ and δ > 0. Statement (1.3) is called Smoluchowski-Kramers µ approximation of qt by qt . This statement justifies the description of the motion of a small particle by the first order equation (1.2) instead of the second order equation (1.1). µ Actually, the closeness of qt and qt is not restricted to equality (1.3). If b(q) is a potential vector, that is b(q) = −∇U (q) for any q ∈ R n , and if σ (q) = I is the unit matrix, then the distribution with the density mµ (q, p) = cµ exp{−(µ|p|2 + 2U (q))} µ

(Boltzman distribution) is invariant for the 2n-dimensional Markov process Xt = µ µ µ µ (qt , pt ), with pt = q˙t , if
 cµ :=

R n ×R n

−1 mµ (q, p) dq dp

> 0.

To prove this one can check that mµ (q, p) satisfies the stationary forward Kolµ mogorov equation associated with the process Xt . µ Then the stationary distribution of qt is equal to  m(q) = mµ (q, p)dp = c˜ exp{−2U (q)}. Rn

On the other hand, m(q) is the stationary density of the process qt defined by (1.2) with σ = I . Thus, if b(q) = −∇U (q) and σ (q) ≡ I , the stationary distributions µ of qt and qt coincide for any µ > 0. This means that, under certain conditions µ providing ergodicity, we can conclude that qt and qt are close not just on finite time intervals, but also have similar long time behavior. If b(x) is not potential, then the invariant measures are not the same. The process µ µ µ Xt = (qt , pt ) may have no finite invariant measure when qt has such a measure. For all details and proofs on this finite dimensional case we refer to [7]. We also refer to [16] and [17] for related problems in finite dimension. In this paper we consider the equation  2  µ ∂∂t 2u (t, x) =
u(t, x) − ∂u  ∂t (t, x) + b(x, u(t, x))   ∂W Q  + ∂t (t, x), t > 0, x ∈ O, (1.4)    ∂u   u(0, x) = u0 , (0, x) = v0 , u(t, x) = 0, x ∈ ∂O. ∂t where O is a bounded open subset of Rd , with d ≥ 1. Here W Q (t, x) is a Gaussian mean zero random field, δ-correlated in time and the operator Q characterizes the correlation in the space variables (see below for detailed assumptions). In particular, in the one-dimensional case W Q (t, x) can be the Brownian sheet, so that ∂2W Q ∂t ∂x (t, x) in this case is the space-time white noise. Together with the semi-linear wave equation with the damping term (1.4), consider the heat equation

On the Smoluchowski-Kramers approximation for a system

  ∂W Q ∂u   (t, x) =
u(t, x) + b(x, u(t, x)) + (t, x), ∂t ∂t    u(0, x) = u0 , u(t, x) = 0, x ∈ ∂O.

365

t > 0, x ∈ O,

(1.5)

Our first result concerns the convergence of uµ (t, x) to u(t, x), as µ ↓ 0. We prove this convergence in Section 4 under some natural assumptions. The proof follows, in general, the arguments used in the finite-dimensional case. But, of course, in the infinite-dimensional case we have to introduce appropriate functional spaces and obtain certain bounds uniform with respect to µ ∈ (0, 1], whose proof requires some work. These auxiliary results together with some notations and assumptions are presented in Sections 2 and 3. The results of these sections allow also to address the questions concerning the invariant measures and the long time behavior for uµ (t, x) and u(t, x). First, we give an explicit (in a sense) expression for the Boltzman distribution of the process µ (uµ (t, x), ∂u ∂t (t, x)). Of course, since there is no universal measure in the functional space similar to the Lebesgue measure, we have to introduce an auxiliary Gaussian measure with respect to which one can write down the density of the Boltzman distribution. This auxiliary Gaussian measure is the stationary measure of the linear wave equation related to problem (1.4). Using the fact that the vector field B[u] := u + b(t, u) in the appropriate functional space is of gradient type, we can express the invariant density through the corresponding potential. µ The explicit expression for the invariant measure of the process (uµ , ∂u ∂t ) allows µ to prove that u (t, x) has the same stationary distributions for each µ > 0, which coincide with the invariant measure of the process u(t, x) defined as the unique solution of the heat equation (1.5). The convergence result of Section 4 will be preserved if we replace the Laplacian  by any second order uniformly elliptic operator with sufficiently smooth coefficients. The results on stationary distributions and invariant measures of Section 5 can be generalized only to self-adjoint non-degenerate second order differential operators with regular coefficients. Actually, if the operator is not self-adjoint, the problem will not be of gradient type. µ,ε Now, let qt and qtε be the solutions of equations (1.1) and (1.2) with σ (x) = εI , 0 < ε 0 and δ ∈ R we define on Hδ the unbounded operator Aµ by setting Aµ (h, k) =

1 (µk,
h − k) , µ

(h, k) ∈ D(Aµ ) := Hδ+1 .

Here for the sake of simplicity we have not written the dependence of Aµ on δ, as the operators Aµ defined on different Hδ are all consistent. It is known that Aµ is the generator of a group of bounded linear transformations {Sµ (t)}t∈ R on Hδ which is strongly continuous (for a proof see e.g. [18, section 7.4]). Note that the adjoint operator to Aµ is given by A µ (h, k) =

1 (−k, −µ
h − k) , µ

(h, k) ∈ D(A µ ) := Hδ+1 .

In what follows we shall denote by {Sµ (t)}{t≥0} the semigroup generated by A µ . Clearly, for any (u0 , v0 ) ∈ Hδ and for any µ > 0, Sµ (t)(u0 , v0 ) is the solution of the deterministic linear system  ∂v ∂u   (t, x) = v(t, x), µ (t, x) =
u(t, x) − v(t, x), t > 0, x ∈ O, ∂t ∂t   u(0) = u0 , v(0) = v0 , u(t, x) = 0, t ≥ 0, x ∈ ∂O,

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which can be written as the following abstract evolution problem in Hδ dz (t) = Aµ z(t), dt

z(0) = (u0 , v0 ),

where z(t) := (u(t), v(t)). Our aim now is giving an explicit expression both of Sµ (t)(u, v) and of Sµ (t) (u, v), for any t ≥ 0 and (u, v) ∈ Hδ . By writing Sµ (t)(u, v) in Fourier coefficients, if we set 1 (u, v) := u and 2 (u, v) := v we have that 1 Sµ (t)(u, v) =

∞ 

µ

fk (t)ek ,

2 Sµ (t)(u, v) =

k=1 µ

∞ 

µ

gk (t)ek ,

k=1

µ

where the pair (fk (t), gk (t)) is for each k ∈ N and µ > 0 the solution of the system    f (t) = g(t), f (0) = uk (2.3)  µ g  (t) = −αk f (t) − g(t), g(0) = vk . µ

µ

In the next proposition we provide an explicit formula for fk and gk . Proposition 2.2. For any µ > 0 and k ∈ N, let us define 1 µ 1 − 4αk µ. γk := 2µ Then, we have µ fk (t)

and µ gk (t)


 

µ  1 t 1 = exp − 1+ µ exp γk t 2 2µ 2µγk



µ  1 + 1− uk µ exp −γk t 2µγk

µ 

µ  1  + µ exp γk t − exp −γk t vk , γk

(2.4)




µ 

µ  t αk  1 − µ exp γk t − exp −γk t uk = exp − 2 2µ µγk 



µ 

µ  1 1 + 1− vk , (2.5) µ exp γk t + 1 + µ exp −γk t 2µγk 2µγk µ

where, in the case γk = 0, we have set

µ 

µ  1  = 2t. µ exp γk t − exp −γk t γk

On the Smoluchowski-Kramers approximation for a system

369

Proof. Differentiating the second equation in system (2.3) we have µ

µ

µ

µ

µ

d 2 gk dg df dg µ (t) = −αk k (t) − k (t) = −αk gk (t) − k (t). 2 dt dt dt dt

Thus, by taking into account the initial conditions, by standard computations we obtain formulas (2.4) and (2.5).   Next, we show that we can express Sµ (t) in terms of Sµ (t). Proposition 2.3. For any µ > 0 and (u, v) ∈ Hδ we have

 Sµ (t)(u, v) = 1 Sµ (t) (u, −v/µ) , 2 Sµ (t) (−µu, v) ,

t ≥ 0. (2.6)

Proof. If we write Sµ (t)(u, v) in Fourier coefficients, we have 1 Sµ (t)(u, v)

=

∞ 

µ fˆk (t)ek ,

2 Sµ (t)(u, v)

=

k=1

is for each k ∈ N and µ > 0 the solution of the

   µf (t) = −g(t), 

µ

gˆ k (t)ek ,

k=1

µ µ (fˆk (t), gˆ k (t))

where the pair system

∞ 

f (0) = uk

µ g  (t) = µ αk f (t) − g(t),

g(0) = vk .

This means that the pair (−µfˆk (t), gˆ k (t)) is the solution of system (2.3) with initial conditions (−µuk , vk ), so that for any t ≥ 0 we have µ

µ

 1 µ fˆk (t) = − 1 Sµ (t)(−µu, v) k , µ

  µ gˆ k (t) = 2 Sµ (t)(−µu, v) k .

This allows us to conclude, as    1 µ 1 Sµ (t)(u, v) k = fˆk (t) = − 1 Sµ (t)(−µu, v) k µ   = 1 Sµ (t) (u, −v/µ) k , and 



2 Sµ (t)(u, v)

k

  µ = gˆ k (t) = 2 Sµ (t) (−µu, v) k .  

Finally, an important consequence of Proposition 2.2 is the following result on the asymptotic behavior of Sµ (t). Proposition 2.4. For any fixed µ > 0 and any δ ∈ R, the semigroup {Sµ (t)}t≥0 is of negative type in Hδ , that is there exist some ωµ > 0 and Mµ > 0 such that Sµ (t) L(Hδ ) ≤ Mµ e−ωµ t ,

t ≥ 0.

(2.7)

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S. Cerrai, M. Freidlin µ

Proof. Fix µ > 0. Multiplying the second equation in (2.3) by gk (t) we get µ

µ

µ

d |gk |2 d |fk |2 µ (t) + αk (t) + 2 |gk (t)|2 = 0, dt dt

and hence, integrating with respect to t ≥ 0 and multiplying both sides by αkδ−1 , we get  t µ µ δ−1 µ δ−1 2 δ 2 µ αk |gk (t)| + αk |fk (t)| + 2 αk |gk (s)|2 ds 0

= µ αkδ−1 |vk |2 + αkδ |uk |2 .

(2.8)

Now, in order to prove (2.7), we note that thanks to Proposition 2.2 for any constant c > 0 and any k ∈ N lim

sup

µ

t→∞ |u |+|v |≤c k k

|fk (t)| = lim

sup

t→∞ |u |+|v |≤c k k

µ

|gk (t)| = 0.

Thus, according to (2.8) we can conclude that for any fixed µ > 0 lim Sµ (t) Hδ = 0.

t→∞

As a consequence of the Datko theorem (see [1] for a proof) this yields (2.7).

 

3. Estimates for the stochastic convolution For each µ > 0, let us consider the linear problem  2 ∂ η ∂W Q ∂η     µ ∂t 2 (t, x) =
η(t, x) − ∂t (t, x) + ∂t (t, x), t > 0, x ∈ O, (3.1)   ∂η   η(0) = 0, (0) = 0, η(t, x) = 0, t ≥ 0, x ∈ ∂O, ∂t where W Q is the noise with covariance given by     E W Q (t), h W Q (s), k = (t ∧ s) Qh, kH . H

Note that

W Q (t)

H

is formally defined as

W Q (t) =

∞ 

Qek βk (t) =

k=1

∞ 

λk ek βk (t),

t ≥ 0,

k=1

where {βk (t)}k∈ N is a sequence of mutually independent standard Brownian motions, all defined on some complete stochastic basis (, F, Ft , P). It is well known that if for some θ ∈ R condition (2.2) holds, then for any µ > 0 there exists a unique solution ηµ to problem (3.1) such that for any T > 0 and p≥1 ∂ηµ ∈ Lp (; C([0, T ]; H θ−1 (O))) (3.2) ∂t (for a proof we refer for example to [4] and [9], see also [3]). ηµ ∈ Lp (; C([0, T ]; H θ (O))),

On the Smoluchowski-Kramers approximation for a system

371

Our aim here is proving that if the constant θ above is strictly positive (as in Hypothesis 1), then for any δ < θ/2 the process ηµ has a version which is δ-H¨older continuous with respect to t ≥ 0 and ξ ∈ O¯ and the momenta of the δ-H¨older norms of ηµ are equi-bounded with respect to µ > 0. Namely we prove the following result. Proposition 3.1. Assume that Hypothesis 1 is satisfied. Then for any µ > 0 and δ < θ/2 the process ηµ has a version (which we still denote by ηµ ) which is ¯ for any T > 0. δ-H¨older continuous with respect to (t, x) ∈ [0, T ] × O, Moreover, for any p ≥ 1 sup E |ηµ | µ>0

p C δ ([0,T ]×O¯ )

=: cT ,p < ∞.

(3.3)

Proof. For all (t, x) ∈ [0, ∞) × O¯ we have ηµ (t, x) =

∞ 

µ

(3.4)

ηk (t)ek (x),

k=1 µ

where, for each k ∈ N, ηk (t) is the solution of the one dimensional problem  µ µ dηk (t) = θk (t) dt     

 µ µ µ µ dθk (t) = − αk ηk (t) + θk (t) dt + λk dβk (t), (3.5)      µ µ ηk (0) = 0, θk (0) = 0. Then, by the variation of constants formula, it is immediate to check that  λk t µ µ f (t − s) dβk (s), ηk (t) = µ 0 k µ

(3.6) µ

with fk defined as the solutions of the system (2.3) with initial conditions fk (0) = µ 0 and gk (0) = 1. Therefore, since for any t, s ≥ 0 and x, y ∈ O¯ the random variable ηµ (t, x) − µ η (s, y) is Gaussian, the proof of (3.3) is a consequence of the following lemma and of the Garcia-Rademich-Rumsey theorem.   Lemma 3.2. Under Hypothesis 1 there exists a constant c > 0 such that   sup E |ηµ (t, x) − ηµ (s, y)|2 ≤ c |t − s|θ + |x − y|2θ ,

(3.7)

µ>0

¯ for any t, s ≥ 0 and x, y ∈ O. Proof. First step. There exists c1 > 0 such that for any t ≥ 0 and x, y ∈ O¯ sup E |ηµ (t, x) − ηµ (t, y)|2 ≤ c1 |x − y|2θ . µ>0

(3.8)

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Due to (3.4) and (3.6), for any t ≥ 0 and x, y ∈ O¯ we have η (t, x) − η (t, y) = µ

µ

 ∞  λk k=1

µ

t 0

µ

fk (t − s) dβk (s) [ek (x) − ek (y)],

so that  t ∞  λ2k µ E |η (t, x) − η (t, y)| = |f (s)|2 ds |ek (x) − ek (y)|2 . µ2 0 k µ

2

µ

k=1

Hence, due to (2.1) E |ηµ (t, x) − ηµ (t, y)|2 ≤ c

∞  λ2 α θ k k µ2

k=1

 |ek |2∞

t 0

µ

|fk (s)|2 ds |x − y|2θ . (3.9)

Now, in order to estimate the series above we can assume µ = 1/4αk , for any k ∈ N. Actually, if we can prove the upper bound  t 1 µ |fk (s)|2 ds =: ck < ∞, sup 2 µ 0 µ=1/4αk since lim

1/4αk

µ

µ→1/4αk

fk (t) = fk

(t),

t ≥ 0,

(3.10)

due to the Fatou lemma we have the same upper bound for any µ > 0. As we are assuming µ = 1/4αk , with a change of variable we have µ E |ηk (t)|2

λ2 = k2 µ



t 0

 µ 2 f (s) ds k

 s  µ µ 2 = exp(γk s) − exp(−γk s) ds exp − µ 2 µ |2µ γk | 0  t 2     µ λk µ = exp − 1 − (1 − 4αk µ)+ s |1 − 4αk µ| 0  µ 2 × 1 − exp(−2µ γ s) ds. λ2k






t

k

If 0 < (1 − 4αk µ)+ ≤ 1/2, we have       1 − exp(−2µ γ µ s)2  s k + exp − 1 − (1 − 4αk µ) s ≤ c exp − s 2 , |1 − 4αk µ| 2 so that µ E |ηk (t)|2

 ≤

c λ2k µ

0

∞

 s exp − s 2 ds = c λ2k µ. 2

On the Smoluchowski-Kramers approximation for a system

Since

  1 − exp(−2µ γ µ s)2 k

|1 − 4αk µ|

if

373

≤ c,

(1 − 4αk µ)+ > 1/2 we have  ∞     µ exp − 1 − (1 − 4αk µ)+ s ds E |ηk (t)|2 ≤ c λ2k µ =

Finally, if



c λ2k



αk

1+



0

 (1 − 4αk µ)+ .

(1 − 4αk µ)+ = 0 we have     1 − exp(−2µ γ µ s)2   k + exp − 1 − (1 − 4αk µ) s |1 − 4αk µ| 1 − cos (1 − 4αk µ)− s = 2 exp(−s) . (1 − 4αk µ)−

Thus, since for any δ ∈ [0, 2] there exists cδ > 0 such that 1 − cos β ≤ cδ for δ = 2 we have

βδ , βδ ∨ 1

β > 0,

(3.11)

 t µ cλ2k 4αk µ s 2 exp (−s) ds αk 0 (1 − 4αk µ)− s 2 ∨ 1  ∞   cλ2k ≤ exp (−s) 1 + s 2 ds. αk 0

µ

E |ηk (t)|2 ≤

Therefore, in all these three cases we obtain µ

E |ηk (t)|2 ≤

cλ2k , αk

(3.12)

and hence, according to (3.9), we obtain (3.8). Second step. There exists a constant c2 > 0 such that for any t, s ≥ 0 and x∈ O sup E |ηµ (t, x) − ηµ (s, x)|2 ≤ c2 |t − s|θ . µ>0

We can assume t > s. As a consequence of (3.6), for any x ∈ O¯ we have
 t ∞  λk µ µ µ η (t, x) − η (s, x) = fk (t − r) dβk (r) µ 0 k=1   s µ − fk (s − r) dβk (r) ek (x) 0

(3.13)

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S. Cerrai, M. Freidlin

=


 ∞  λk k=1



µ

+

s

t s

0

µ fk (t

µ

fk (t − r) dβk (r) µ − r) − fk (s





− r) dβk (r) ek (x),

and hence we have  2 E ηµ (t, x) − ηµ (s, x)  ∞  λ2k t µ = |f (t − r)|2 dr |ek |2∞ µ2 s k k=1  ∞  2 λ2k s  µ µ f (t − r) − fk (s − r) dr |ek |2∞ + µ2 0 k k=1

=:

∞ 

µ

Ik |ek |2∞ +

k=1

∞ 

µ

Jk |ek |2∞ .

(3.14)

k=1 µ

Concerning the terms Ik , since for any δ ∈ [0, 1] there exists cδ > 0 such that 1 − exp(−β) ≤ cδ β δ ,

β > 0,

due to (3.12) we have  λ2 t−s µ µ Ik = k2 |fk (r)|2 dr µ 0  



cλ2k 1 − (1 − 4αk µ)+ ≤ (t − s) 1 − exp − αk 2µ

θ cλ2k cλ2k 1 − (1 − 4αk µ)+ ≤ (t − s)θ ≤ 1−θ (t − s)θ . αk 2µ(1 + (1 − 4αk µ)+ ) αk µ

(3.15)

(3.16)

Now we go to the estimate of the terms Jk , which is more delicate. As in the first step, due to (3.10) we can assume that 4αk µ = 1. We have  µ µ µ 2γk fk (t − r) − fk (s − r)
 s−r  µ = exp − exp(γk (s − r)) 2µ 

 µ  (1 − 2γk µ) µ − exp(−γk (s − r)) exp − (t − s) − 1 2µ
 t −s  µ + exp − exp(γk (t − s)) 2µ

µ  (1 + 2γk µ) µ − exp(−γk (t − s)) exp − (s − r) . 2µ

On the Smoluchowski-Kramers approximation for a system

375

This implies that µ Jk

≤



cλ2k

µ |2γk µ|2

s 0




s−r exp − µ



  exp(γ µ (s − r)) − exp(−γ µ (s − r))2 dr k k

 2

µ   (1 − 2γk µ)   × exp − (t − s) − 1   2µ     (1 + 2γ µ µ)  s cλ2k   k + exp − (s − r) dr µ 2   µ |2γk µ| 0
 2 t − s  µ µ µ µ × exp − exp(γk (t − s)) − exp(−γk (t − s)) =: (Jk )1 + (Jk )2 . µ µ

µ

We estimate separately the terms (Jk )1 and (Jk )2 . Since 

λ2k µ

|2γk µ|2

s 0




 r  µ µ 2 µ exp − exp(γk r) − exp(−γk r) dr = E |ηk (s)|2 , µ

according to (3.12) we have µ (Jk )1

cλ2 ≤ k αk



2 µ   (1 − 2γk µ)   (t − s)  . 1 − exp −   2µ

It is not difficult to check that if z ∈ C |1 − exp(−z)|2 = (1 − exp(−Re z))2 + 2 exp(−Re z) (1 − cos Im z) . (3.17) Then we have 

2 µ   (1 − 2γk µ)   (t − s)  1 − exp −   2µ 



2 1 − (1 − 4αk µ)+ ≤ 1 − exp − (t − s) 2µ

(1 − 4αk µ)− +2 1 − cos (t − s) . 2µ Then, from (3.15) and (3.11) it follows 

θ θ  2 + − cλ 1 − (1 − 4α µ) (1 − 4α µ) k k µ  (t − s)θ + (Jk )1 ≤ k  αk 2µ 2µ ≤

cλ2k αk1−θ

(t − s)θ .

(3.18)

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S. Cerrai, M. Freidlin µ

Concerning (Jk )2 we have µ (Jk )2


µ t − s  µ exp(γk (t − s))− ≤ µ µ exp − µ |2γk µ|2 1 + Re 2γk µ 2 µ exp(−γ (t − s)) cλ2k

k

=

cλ2k µ |1 − 4αk µ|(1 + (1 − 4αk µ)+ )



1 − (1 − 4αk µ)+ exp − (t − s) µ  2

µ × 1 − exp −2γk (t − s)  .

Then, due to (3.17), (3.15) and (3.11) for any δ ∈ [0, 2] we have

cδ λ2k µ1−δ 1− (1−4αk µ)+ µ exp − (t−s) (t−s)δ (Jk )2 ≤ + µ 1+ (1−4αk µ)  δ/2 (1 − 4αk µ)+ × |1 − 4αk µ| −1  δ/2


δ/2 − (1−4αk µ)− (1−4αk µ)  . (3.19) + (t−s)δ ∨ 1 |1−4αk µ| µ2 If we take δ = θ in (3.19) and assume 4αk µ ∈ (0, 1/2], we have µ

(Jk )2 ≤

cθ λ2k µ1−θ c λ2k θ (t − s) ≤ (t − s)θ . (1 − 4αk µ)1−θ/2 αk1−θ

(3.20)

If we take δ = 2 in (3.19) and assume 4αk µ ∈ (1/2, 1), we have √ 
cλ2 (1 − 1 − 4αk µ) µ (t − s) . (Jk )2 ≤ (t − s)2 exp − √ k µ µ(1 + 1 − 4αk µ) Now, we remark that for any β > 0 there exists a constant cβ > 0 such that s β e−s ≤ cβ ,

s ≥ 0,

(3.21)

so that √ √ 
2−θ
1 + 1 − 4αk µ (1 − 1 − 4αk µ) 2−θ (t − s) (t − s) ≤c µ2−θ . exp − µ 4αk µ As 4αk µ > 1/2 this yields µ

(Jk )2 ≤

cλ2 (t − s)θ √ k µ(1 + 1 − 4αk µ)

≤ cλ2k µ1−θ (t − s)θ , and hence, as 4αk µ < 1, (3.20) follows.




1+

√

1 − 4αk µ 4αk µ

2−θ µ2−θ

On the Smoluchowski-Kramers approximation for a system

377

Next, if we take δ = 2 in (3.19) and assume 4αk µ ∈ (1, 2), due to (3.21) we get µ (Jk )2


 cλ2k cλ2k t −s ≤ (t − s)θ . exp − (t − s)2 ≤ 1−θ µ µ αk

Finally, if we assume 4αk µ ≥ 2, by taking again δ = 2 in (3.19) we have
−1
 cλ2k t −s µ 2 (4αk µ − 1) 2 (Jk )2 ≤ (t − s) ∨ 1 exp − (t − s) µ µ µ2 ≤

cλ2k µ1−θ cλ2k θ (t − s) ≤ (t − s)θ , (αk µ)1−θ/2 αk1−θ

so that (3.20) holds. According to (3.14), thanks to (3.18) and (3.20) we obtain (3.13). Conclusion. Estimate (3.7) follows combining together (3.8) and (3.13).

 

4. The convergence result In this section we are concerned with the stochastic semi-linear damped wave equation  2  µ ∂∂t 2u (t, x) =
u(t, x) − ∂u  ∂t (t, x) + b(x, u(t, x))   ∂W Q  + ∂t (t, x), t > 0, x ∈ O, (4.1)    ∂u   u(0) = u0 , u(t, x) = 0, t ≥ 0, x ∈ ∂O. (0) = v0 , ∂t Our aim is proving that the solution uµ (t) converges to the solution of the stochastic semi-linear heat equation  ∂z  ∂t (t, x) =
z(t, x) + b(x, z(t, x)) Q (4.2) t > 0, x ∈ O, + ∂W ∂t (t, x),  z(0) = u0 , z(t, x) = 0, t ≥ 0, x ∈ ∂O, as the parameter µ converges to zero. For any µ > 0 and δ ∈ [0, 1] we define the operators Bµ (h, k)(x) :=

1 (0, b(x, h(x))), (h, k) ∈ Hδ , x ∈ O, µ

(4.3)

and Qµ h =

1 (0, Qh), µ

h ∈ H.

(4.4)

Note that, since δ ∈ [0, 1], for any z1 = (u1 , v1 ) and z2 = (u2 , v2 ) ∈ Hδ |Bµ (z1 ) − Bµ (z2 )|Hδ =

1 c |b(·, u1 ) − b(·, u2 )|H δ−1 (O) ≤ |b(·, u1 ) − b(·, u2 )|H , µ µ

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and then, thanks to Hypothesis 2 |Bµ (z1 ) − Bµ (z2 )|Hδ ≤

cL cL |u1 − u2 |H ≤ |z1 − z2 |Hδ . µ µ

(4.5)

Definition 4.1. Let δ ∈ [0, 1]. A process uµ is a mild solution of problem (4.1) if uµ ∈ L2 (; C([0, T ]; H δ (O))),

v µ :=

∂uµ ∈ L2 (; C([0, T ]; H δ−1 (O))), ∂t

for any T > 0, and 

t

z (t) = Sµ (t)(u0 , v0 ) + µ



t

Sµ (t − s)Bµ (z (s)) ds + µ

0

Sµ (t − s) dW Qµ (s),

0

where zµ (t) := (uµ (t), v µ (t)). Note that with these notations, the weak solution ηµ of problem (3.1) introduced in Section 3 can be written as  t ηµ (t) = 1 Sµ (t − s) dW Qµ (s), t ≥ 0, 0

and hence if uµ is a mild solution of problem (4.1) we have uµ (t) = 1 Sµ (t)(u0 , v0 )  t +1 Sµ (t − s)Bµ (uµ (s), v µ (s)) ds + ηµ (t),

t ≥ 0. (4.6)

0

Now we can establish the existence of a unique mild solution to problem (4.1), for any µ > 0. This result is known in the literature (for a proof see e.g. [5, Theorem 5.3.1]), but here, for the safe of completeness, we give a self-contained proof. Proposition 4.2. Assume Hypotheses 1 and 2. Then for any µ > 0 and for any initial data u0 ∈ H θ (O) and v0 ∈ H θ−1 (O) there exists a unique mild solution zµ (t) to problem (4.1). Moreover, for any T > 0 and p ≥ 1 there exists cµ,p (T ) > 0 such that   p p E sup |zµ (t)|Hθ ≤ cµ,p (T ) 1 + |(u0 , v0 )|Hθ . (4.7) t∈ [0,T ]

Proof. For any z = (u, v) ∈ L2 (; C([0, T ]; H θ (O))) × L2 (; C([0, T ]; H θ−1 (O))) =: Hθ (T ) we define  t  t Fµ (z)(t) := Sµ (t)(u0 , v0 ) + Sµ (t − s)Bµ (z(s)) ds + Sµ (t − s) dW Qµ (s). 0

0

If we show that for some T0 > 0 small enough Fµ is a contraction on Hθ (T0 ), then we have a unique mild solution to problem (4.1) in the interval [0, T0 ]. Due to (2.7), we have Sµ (t)(u0 , v0 ) ∈ Hθ (T ),

T > 0.

On the Smoluchowski-Kramers approximation for a system

379

Moreover, as seen in (4.5) Bµ maps Hθ into itself, so that, by using again (2.7) we have  t t → Sµ (t − s)Bµ (z(s)) ds ∈ Hθ (T ). 0

Thanks to (3.2) we can conclude that Fµ (z) ∈ Hθ (T ), for any z ∈ Hθ (T ). Now, thanks to (4.5) Fµ is a contraction on Hθ (T0 ), provided T0 is sufficiently small. This means that Fµ admits a unique fixed point in Hθ (T0 ), which is the unique mild solution defined in the time interval [0, T0 ]. As the same arguments can be repeated in the intervals [T0 , 2T0 ], [2T0 , 3T0 ] and so on, we obtain a unique global solution in the time interval [0, T ]. We skip here the proof of estimate (4.7). A proof can be found for example in [5, Theorem 5.3.1].   Next step is proving that the family of probability measures {L(uµ )}µ∈ (0,1] is tight on C([0, T ]; H ), for any T > 0. Proposition 4.3. Assume that u0 ∈ H 1 (O) and v0 ∈ H . Then, under Hypotheses 1 and 2 the family of probability measures {L(uµ )}µ∈ (0,1] is tight on C([0, T ]; H ), for any T > 0. Proof. If ηµ is the solution of the stochastic linear damped wave equation (3.1) and if we define ρ µ (t) := uµ (t) − ηµ (t),

t ≥ 0,

ρ µ (t)

then the process solves the deterministic equation  2 µ ∂ ρ ∂ρ µ   µ ∂t 2 (t, x) =
ρ µ (t, x) − ∂t (t, x) + b(x, ρ µ (t, x) +ηµ (t, x)), t > 0, x ∈ O,  µ  µ ρ (0) = u0 , ∂ρ∂t (0) = v0 , ρ µ (t, x) = 0, t ≥ 0, x ∈ ∂O. If we multiply both sides of the first equation above by (−
)θ−1 ∂ρ µ /∂t and integrate with respect to x ∈ O, we easily get    µ 2   ∂ρ d  ∂ρ µ 2 d  µ 2 µ (t) ρ (t) H θ (O) + 2  (t) +  dt ∂t dt ∂t H θ −1 (O ) H θ −1 (O )   µ ∂ρ = 2 b(·, ρ µ (t) + ηµ (t)), (−
)θ−1 (t) ∂t H  µ 2   ∂ρ ≤ |b(·, ρ µ (t) + ηµ (t))|2H +  (t) . ∂t θ −1 H

(O )

Hence, integrating with respect to t ≥ 0 it follows  µ 2  t  µ 2  ∂ρ    ∂ρ  2  µ  (t) (s) + ρ µ (t)H θ (O) + ds  ∂t ∂t 0 H θ −1 (O ) H θ −1 (O )  t 2 2 ≤ µ |v0 |H θ −1 (O) + |u0 |H θ (O) + |b(·, ρ µ (s) + ηµ (s))|2H ds. 0

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Now, due to Hypothesis 2, for any s ∈ [0, T ] we have



|b(·, ρ µ (s) + ηµ (s))|2H ≤ c 1 + |ρ µ (s)|2L2 (O) + sup |ηµ (t)|2H

,

t∈ [0,T ]

and then  µ 2  t  µ 2  ∂ρ  ∂ρ  µ 2        µ + ρ (t) H θ (O) + ds (t) (s)  ∂t ∂t 0 H θ −1 (O ) H θ −1 (O ) ≤ µ |v0 |2H θ −1 (O) + |u0 |2H θ (O)

+cT

1 + sup |ηµ (t)|2H t∈ [0,T ]

 +c

t

0

|ρ µ (s)|2H θ (O) ds.

Thanks to the Gronwall lemma, for any µ ∈ (0, 1] and T > 0 this yields  T  µ 2   ∂ρ  µ 2     sup ρ (t) H θ (O) + ds  ∂t (s) θ −1 0 t∈ [0,T ] H (O )

≤ cT

|v0 |2H θ −1 (O) + |u0 |2H θ (O) + sup |ηµ (t)|2H + 1 .

(4.8)

t∈ [0,T ]

According to (3.3), this means that we can find some constant cT independent of µ ∈ (0, 1] such that  T  µ 2  ∂ρ  2    ds ≤ cT . E sup ρ µ (t)H θ (O) + E  ∂t (s) θ −1 0 t∈ [0,T ] H (O ) In particular ρ µ ∈ L2 (; C([0, T ]; H θ (O))) and, since uµ = ρ µ + ηµ , due to (3.3) we have that uµ ∈ L2 (; C([0, T ]; H )) and sup E |uµ |2C([0,T ];H ) ≤ cT .

(4.9)

µ∈ (0,1]

Next, by using once more (3.3), for any  > 0 we can find λ > 0 such that 

(4.10) P ηµ ∈ Kλ,1 ≥ 1 − , µ > 0, where, by the Ascoli-Arzel`a theorem, Kλ,1 is the compact subset of C([0, T ]; H ) defined by   Kλ,1 := f : [0, T ] × O¯ → R, : |f |C δ ([0,T ]×O¯ ) ≤ λ , with δ < θ/2. Now, as we are assuming that u0 ∈ (4.8) we have  2  ηµ ∈ Kλ,1 ⇒ sup ρ µ (t)H 1 (O) + sup t∈ [0,T ]

t∈ [0,T ] 0

H 1 (O) and v0 ∈ H , due to t

 µ 2  ∂ρ     ∂t (s) ds ≤ cT ,λ , H

On the Smoluchowski-Kramers approximation for a system

381

so that  ηµ ∈ Kλ,1 ⊂ uµ = ρ µ + ηµ ∈ Kλ,2 + Kλ,1 ,



where, again by the Ascoli-Arzel`a theorem, Kλ,2 is the compact subset of C([0, T ]; H ) defined by     |f (t) − f (s)|H 1/2 ≤ c Kλ,2 := f : sup |f (t)|H 1 (O) ≤ cT ,λ , sup T ,λ  .  |t − s|1/2 t,s∈ [0,T ] t∈ [0,T ] t=s

Hence, in view of (4.10) we have 

P uµ ∈ Kλ,1 + Kλ,2 ≥ 1 − , and this concludes the proof of tightness.

 

Next, we prove an integration by parts formula. Lemma 4.4. Assume Hypotheses 1 and 2 and fix u0 ∈ H θ (O) and v0 ∈ H θ−1 (O). ¯ such that ϕ ≡ 0 on ∂O, we Then for any µ > 0 and for any ϕ ∈ C 2 ([0, T ] × O), have   uµ (t, x)ϕ(t, x) dx = u0 (x)ϕ(0, x) dx O O $ %  t ∂ϕ + uµ (s, x) (s, x) +
ϕ(s, x) ds dx ∂t 0 O  t + b(x, uµ (s, x))ϕ(s, x) ds dx 0 O  t + ϕ(s, x) W Q (ds, dx) + Rµ (t), (4.11) 0

O

where  t   t  t−s v0 (x)ϕ(0, x) dx − e− µ Mµ (s) ds Rµ (t) : = µ 1 − e− µ 0 $ O
 t ∂ϕ ∂ϕ − t−s − u0 (x) (0, x) − uµ (s, x) (s, x) e µ ∂t ∂t O 0  %  s 2 ∂ ϕ + uµ (r, x) 2 (r, x) dr dx ds ∂t 0  t t−s − e− µ ϕ(s, x)W Q (ds, dx), (4.12) O

0

and

 Mµ (t) :=

O

 uµ (t, x)
ϕ(t, x) + b(x, uµ (t, x))ϕ(t, x) dx.

(4.13)

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S. Cerrai, M. Freidlin

Proof. Since we are assuming u0 ∈ H θ (O) and v0 ∈ H θ−1 (O), due to Proposition 4.2 we only have uµ ∈ L2 (; C([0, T ]; H θ (O))),

∂uµ ∈ L2 (; C([0, T ]; H θ−1 (O))). ∂t

Thus, in order to have enough regularity, in our computations we replace uµ with ¯ its finite dimensional Galerkin approximations which belong to C 2,2 ([0, T ] × O). Once we get formula (4.11) for the Galerkin approximations, we pass easily to the limit and we get formula (4.11) for uµ . Note that for simplicity of notations we still denote the Galerkin approximations by uµ . If we set v µ := ∂uµ /∂t, multiplying both sides of the first equation in ¯ and integrating with respect to t ≥ 0 problem (4.1) by some ϕ ∈ C 2 ([0, T ] × O) and x ∈ O, we get  t

  1 t ∂v µ (s, x)ϕ(s, x) ds dx = ϕ(s, x) W Q (ds, dx) µ 0 O 0 O ∂t     µ 1 t +
u (s, x) − v µ (s, x) + b(x, uµ (s, x)) ϕ(s, x) ds dx. µ 0 O

Now, integrating by parts we have  t 0

& ' ∂v µ (s, x)ϕ(s, x) ds dx = v µ (t), ϕ(t) H − v0 , ϕ(0)H ∂t O   t ∂ϕ µ v (s), (s) − ds, ∂t 0 H

and if ϕ vanishes at the boundary of O  t 0



t


u (s, x)ϕ(s, x) ds dx = µ

O

0

' uµ (s),
ϕ(s) H ds.

&

Thus, if we define 

H (t) : =

&

' & ' v µ (s), ϕ(s) H ds = uµ (t), ϕ(t) H 0   t ∂ϕ uµ (s), ds, (s) − u0 , ϕ(0)H − ∂t 0 H t

and if Mµ (t) is defined as in (4.13), we obtain   1 1 t H  (t) = − H (t) + v0 , ϕ(0)H + ϕ(s), dW Q (s) H µ µ 0  %   t$ 1 ∂ϕ Mµ (s) + µ v µ (s), ds. (s) + µ 0 ∂t H

(4.14)

On the Smoluchowski-Kramers approximation for a system

383

As H (0) = 0, this yields   s   t  1 t − t−s H (t) = µ 1 − e− µ v0 , ϕ(0)H + ϕ(r), dW Q (r) ds e µ H µ 0 0   %  s $  t t−s 1 ∂ϕ + Mµ (r) + µ v µ (r), dr ds. e− µ (r) µ 0 ∂t 0 H Hence, due to (4.14) we get   t ' ∂ϕ µ u (t), ϕ(t) H = u0 , ϕ(0)H + u (s), (s) ds ∂t 0 H   t +µ 1 − e− µ v0 , ϕ(0)H  s   1 t − t−s + e µ (4.15) ϕ(r), dW Q (r) ds H µ 0 0   %  s$  1 t − t−s ∂ϕ + Mµ (r) + µ v µ (r), dr ds. e µ (r) µ 0 ∂t 0 H

&

µ

Now, integrating by parts it is immediate to check that for any function f : [0, T ] → R   s  t  t t−s 1 t − t−s e µ f (r) dr ds = f (s) ds − e− µ f (s) ds. (4.16) µ 0 0 0 0 Similarly, for the stochastic integral we have 1 µ

 s  t−s ϕ(r), dW Q (r) ds e− µ H 0 0  t  t   t−s  = e− µ ϕ(s), dW Q (ds) . ϕ(s), dW Q (s) − 

t

0

H

0

H

(4.17)

Therefore, recalling how Mµ (t) is defined, if we plug (4.16) and (4.17) into (4.15) we obtain   t & µ ' ∂ϕ µ u (t), ϕ(t) H = u0 , ϕ(0)H + u (s), ds (s) ∂t 0 H   t +µ 1 − e− µ v0 , ϕ(0)H  t  t 

& µ ' ϕ(s), dW Q (s) + u (s),
ϕ(s) H + H 0 &0 '  µ + b(·, u (s)), ϕ(s) H ds  t  t  t−s  t−s − e− µ ϕ(s), dW Q (s) − e− µ Mµ (s) ds H 0  0  t  s t−s ∂ϕ − µ µ + e v (r), (r) dr ds. ∂t 0 0 H

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S. Cerrai, M. Freidlin

This concludes the proof of the lemma, as
    t  s  t ∂ϕ ∂ϕ − t−s − t−s µ µ µ µ v (r), u (s), e dr ds = e (r) (s) ∂t ∂t 0 0 0 H H   ∂ϕ (0) − u0 , ∂t H     s 2ϕ ∂ µ u (r), 2 (r) − dr ds. ∂t 0 H   Concerning the remainder term Rµ (t) defined in (4.12), we have the following limiting result. Lemma 4.5. Under the same hypotheses of Lemma 4.4 we have lim E |Rµ (t)|2 = 0,

µ→0

t ≥ 0.

Proof. We have  t  2   2 − t−s Q   µ ϕ(s), dW (s)  |Rµ (t)| ≤ 3 µ v0 , ϕ(0)H + 3  e H 0  t $      ∂ϕ ∂ϕ − t−s µ  µ +3  e Mµ (s) + u0 , (0) − u (s), (s) ∂t ∂t 0 H H  % 2  s 2  ∂ ϕ uµ (r), 2 (r) + dr ds  =: Iµ1 (t) + Iµ2 (t) + Iµ3 (t). ∂t 0 H 

2

2

Now, with a change of variable we have  t  − 2(t−s) 2 2 µ E Iµ (t) = 3 e |ϕ(s)|H ds = 3 µ

0

0

≤

t µ

e−2s |ϕ(t − µs)|2H ds

3 µ sup |ϕ(s)|2H 2 s∈ [0,T ]

Moreover, recalling how Mµ (t) is defined in (4.13), with a change of variables we easily get  ∞ e−2s ds (1 + T 2 )|uµ |2C([0,T ];H ) |ϕ|2C 2 ([0,T ]×O¯ ) , Iµ3 (t) ≤ µ c t 0

and hence, due to (4.9) we conclude   E Iµ3 (t) ≤ µ cT 1 + E |uµ |2C([0,T ];H ) |ϕ|2C 2 ([0,T ]×O¯ ) ≤ µ cT |ϕ|2C 2 ([0,T ]×O¯ ) . This implies that E |Rµ (t)|2 ≤ Iµ1 (t) + E Iµ2 (t) + E Iµ3 (t) ≤ µ cT , for some constant cT depending only on T , u0 and ϕ, and the lemma is proved.  

On the Smoluchowski-Kramers approximation for a system

385

Now we can prove the main result of this section. Theorem 4.6. Assume Hypotheses 1 and 2 and fix u0 ∈ H 1 (O) and v0 ∈ H . Then, if uµ is the solution of the stochastic semi-linear damped wave equation (4.1) and z is the solution of the stochastic semi-linear heat equation (4.2), for any T > 0 and for any  > 0 we have

 lim P |uµ − z|C([0,T ];H ) >  = 0. µ→0

Proof. Due to the tightness in C([0, T ]; H ) of the sequence {L(uµ )}µ∈ (0,1] , the Skorokhod theorem assures that for any two sequences {µn }n and {µm }m converging to zero there exist subsequences {µn(k) }k∈ N and {µm(k) }k∈ N and a sequence of random elements   Q {ρk }k∈ N := (uk1 , uk2 , Wˆ k ) , k∈ N

C([0, T ]; D (O)),

× defined on some probability space in C := ˆ such that the law of ρk coincides with the law of (uµn(k) , uµm(k) , W Q ), for ˆ P), ˆ F, (, ˆ to some random element ρ := (u1 , u2 , Wˆ Q ) ∈ each k ∈ N, and ρk converges P-a.s. C. Now, if we show that u1 = u2 , we have that there exists some z ∈ C([0, T ]; H ) such that uµ converges to z in probability. Actually, as observed by Gy¨ongy and Krylov in [8], if E is any Polish space equipped with the Borel σ -algebra, a sequence {ρn } of E-valued random variables converges in probability if and only if for every pair of subsequences {ρm } and {ρl } there exists an E 2 -valued subsequence wk := (ρm(k) , ρl(k) ) converging weakly to a random variable w supported on the diagonal {(h, k) ∈ E 2 : h = k}. Q Note that both uk1 and uk2 solve equation (4.1) with W Q replaced by Wˆ k . Then they both verify formula (4.11), with R1k and R2k obtained replacing uµ respectively Q with uk1 and uk2 and W Q with Wˆ k . According to Lemma 4.5 we have that both R1k ˆ as µn(k) and µm(k) go to zero, and then, possibly and R2k converge to zero in L2 (), ˆ for a subsequence, they converge P-a.s. to zero. Due to formula (4.11) this implies   ui (t, x)ϕ(t, x) dx = u0 (x)ϕ(0, x) dx O O $ %  t ∂ϕ + ui (t, x) (s, x) +
ϕ(s, x) ds dx ∂t 0 O  t + b(x, ui (s, x))ϕ(s, x) ds dx 0 O  t + ϕ(s, x) Wˆ Q (ds, dx), i = 1, 2, C([0, T ]; H )2

0

O

and then both u1 and u2 coincide with the solution of the semi-linear heat equation perturbed by the noise Wˆ Q , which is unique. As we have recalled above, thanks to the remark by Gy¨ongy-Krylov in [8] this implies that uµ converges in probability to some random variable z ∈ C([0, T ]; H ).

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But, by using again formula (4.11) and Lemma 4.5 we have that z solves the heat equation (4.2). This completes the proof of the theorem.   5. Stationary distributions In this section we study the relation between the stationary distributions of the processes uµ (t) and z(t), defined respectively as the solution of the semi-linear stochastic damped wave equation (4.1) and as the solution of the semi-linear stochastic heat equation (4.2). If we set zµ (t) := (uµ (t), v µ (t)),

t ≥ 0, µ > 0,

with the notations introduced in Section 2 and Section 4 we can write equation (4.1) as the abstract evolution equation on the Hilbert space H0 = H × H −1 (O)   dzµ (t) = Aµ zµ (t) + Bµ (zµ (t)) dt + dW Qµ , zµ (0) = (u0 , v0 ), (5.1) where Bµ and Qµ are the operators already defined in (4.3) and (4.4), respectively. Note that the adjoint of the operator Qµ : H → H0 is the operator Q µ : H0 → H defined by Q µ (u, v) =

1 (−
)−1 Qv. µ

In particular we have that Qµ Q µ : H0 → H0 is given by Qµ Q µ (u, v) =

1 (0, (−
)−1 Q2 v), µ2

(u, v) ∈ H0 .

(5.2)

Next, for any µ > 0 we introduce the operator Cµ ∈ L+ (H0 ) by setting 

∞

Cµ := 0

Sµ (s) Qµ Q µ Sµ (s) ds,

where {Sµ (t)}t≥0 is the semigroup generated by A µ , the adjoint to the operator Aµ . Proposition 5.1. Under Hypothesis 1, with θ = 0, we have
 1 1 Cµ (u, v) = (−
)−1 Q2 u, (−
)−1 Q2 v , (u, v) ∈ H0 . 2 µ In particular Cµ is a trace-class operator with Tr Cµ =


 ∞ 1  λ2k 1 1+ . 2 µ αk k=1

(5.3)

On the Smoluchowski-Kramers approximation for a system

387

Proof. Due to (2.6) and (5.2), for any (u, v) ∈ H0 and µ > 0 we have 1 Sµ (t)(0, (−
)−1 Q2 2 Sµ (t)(u, v)) µ2 1 = 2 Sµ (t)(0, (−
)−1 Q2 2 Sµ (t)(−µu, v)). µ

Sµ (t) Qµ Q µ Sµ (t)(u, v) =

Thus, from (2.4) and (2.5), for any k ∈ N we easily have 



1 Sµ (t) Qµ Q µ Sµ (t)(u, v)

k



µ  t αk  = µ exp − µ exp γk t µ 4αk µ2 γk γk 

µ 

µ 2 uk + exp γk t − exp −γk t 

µ 

µ  1 − exp −γk t 1− µ exp γk t 2µγk



µ  1 + 1+ vk , µ exp −γk t 2µγk λ2k




µ

with the usual assumption that if γk = 0 then for any t ≥ 0

µ 

µ  1  = 2t. µ exp γk t − exp −γk t γk Therefore, by some computations for any k ∈ N we obtain  ∞     λ2 1 Cµ (u, v) k = 1 Sµ (t) Qµ Q µ Sµ (t)(u, v) k dt = k uk . 2αk 0 Concerning the second component, due to (2.5) we have 
  λ2k t 2 Sµ (t) Qµ Q µ Sµ (t)(u, v) k = exp − 4αk µ2 µ 





µ 

µ  1 1 1− µ exp γk t + 1 + µ exp −γk t 2µγk 2µγk

µ 

µ  αk  uk µ exp γk t − exp −γk t γk 



µ 

µ  1 1 + 1− vk , µ exp γk t + 1 + µ exp −γk t 2µγk 2µγk and, as for the first component, by some computations this implies that  ∞     λ2k 2 Cµ (u, v) k = 2 Sµ (t) Qµ Q µ Sµ (t)(u, v) k dt = vk . 2µαk 0 This allows to obtain (5.3) and hence to conclude the proof of the proposition.  

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5.1. The linear case Our aim here is studying the invariant measure of the system dz(t) = Aµ z(t) dt + d W Qµ (t),

z(0) = (u0 , v0 ) ∈ H0 ,

(5.4)

and showing that the stationary distribution for the solution of the linear stochastic damped wave equation µ

∂W Q ∂u ∂ 2u (t, x) + (t, x), (t, x) =
u(t, x) − ∂t 2 ∂t ∂t u(t, x) = 0, x ∈ ∂O,

(5.5)

coincides for all µ > 0 with the unique invariant measure of the linear stochastic heat equation ∂W Q ∂u (t, x) =
u(t, x) + (t, x), ∂t ∂t

u(0, x) = 0, x ∈ ∂O.

(5.6)

Theorem 5.2. Under Hypothesis 1, with θ = 0, the Gaussian measure N (0, Cµ ) is the unique invariant measure of system (5.4), for each µ > 0, and for any ϕ ∈ Cb (H0 ) and z0 ∈ H0  lim Ez0 ϕ(zµ (t)) = ϕ(z) N (0, Cµ )(dz), (5.7) t→∞

H0

so that N (0, Cµ ) is ergodic and strongly mixing. Moreover the Gaussian measure ν = N (0, (−
)−1 /2) is the stationary distribution of (5.5). In particular, ν does not depend on µ > 0 and coincides with the unique invariant measure of the stochastic heat equation (5.6). Proof. According to Proposition 5.1, the operator Cµ is non-negative, symmetric and of trace-class on H0 . Thus problem (5.4) admits an invariant measure of the form ν N (0, Cµ ), where ν is an invariant measure for the semigroup Sµ (t) and N (0, Cµ ) is the Gaussian measure, with zero mean and covariance operator Cµ (for a proof see e.g. [4, Theorem 11.7]). Moreover, as the semigroup {Sµ (t)}t≥0 is of negative type (see Proposition 2.4), due to [4, Theorem 11.11] N (0, Cµ ) is the unique invariant measure for (5.1) and (5.7) holds. As well known this implies that N (0, Cµ ) is ergodic and strongly mixing. Next, due to (5.3) the measure N (0, Cµ ) defined on H0 is the product of two Gaussian measures, defined respectively on H and H −1 (O). Namely     N (0, Cµ ) = N 0, (−
)−1 /2 × N 0, (−
)−1 /2µ . In particular the marginal measure νµ := 1 N (0, Cµ ) equals N (0, (−
)−1 /2), so that it does not depend on µ > 0 and coincides with the unique invariant measure ν of the Ornstein-Uhlenbeck process solving problem (5.6).

On the Smoluchowski-Kramers approximation for a system

389

This allows us to conclude the proof of the theorem, as the process u¯ µ (t) = 1 z¯ µ (t), with  t z¯ µ (t) = (u¯ µ (t), v¯ µ (t)) := Sµ (t − s)dW Qµ (s), 0

is the stationary solution to problem (5.5) and its distribution is 1 N (0, Cµ ).

 

5.2. The semi-linear case We show that an analogous result holds also in the non-linear case, if Q is the identity operator (and in particular if d = 1). Our aim first is proving that system (5.1) is of gradient type and admits an invariant measure of the following type νµ (dz) = cµ e 2U (z) N (0, Cµ )(dz), for some mapping U : H0 → R which does not depend on µ > 0 and is a function of u ∈ H only. To this purpose we introduce some notations. For any n ∈ N and ξ = (ξ1 , . . . , ξn ) ∈ R n we define  Tn ξ := ξk e k . k≤n

Clearly the mapping Tn is well defined from Rn into H δ (O) and the mapping T¯n (ξ, η) := (Tn ξ, Tn η) is well defined from Rn × Rn into Hδ , for any δ ∈ R. Moreover, if we define Rn u := ( u, e1 H , . . . , u, en H ) , we have that Rn maps H δ (O) into Rn , for any δ ∈ R, and Rn Tn = IdRn . Furthermore, if we set Pn := Tn Rn , we have that Pn is the projection of H δ (O) onto the finite dimensional space generated by {e1 , . . . , en } and for any fixed u ∈ H δ (O) we have that Pn u converges to u in H δ , as n goes to infinity. In particular, setting P¯n (z) := (Pn u, Pn v),

z = (u, v) ∈ Hδ ,

we have lim P¯n z = z,

n→∞

in Hδ

(5.8)

In what follows, for any Banach space X we denote by Bb (X) the Banach space of Borel and bounded functions from X into R, endowed with the sup-norm, and we denote by Cb (X) the subspace of uniformly continuous functions. We recall that the transition semigroup {P µ (t)}t≥0 associated with system (5.1) in H0 is defined for any t ≥ 0 and ϕ ∈ Bb (H0 ) by P µ (t)ϕ(z) = E ϕ(uµ (t), v µ (t)),

z = (u, v) ∈ H0 ,

where (uµ (t), v µ (t)) is the solution to (5.1) with initial datum z = (u, v).

390

S. Cerrai, M. Freidlin µ

Next, we denote by zn (t) the solution to the finite dimensional problem   dz(t) = Aµ z(t) + P¯n Bµ (P¯n z(t)) dt + dW Iµ,n (t), z(0) = P¯n z, (5.9) where Iµ,n = P¯n Iµ . Due to (5.8), to the fact that Bµ is Lipschitz continuous (see (4.5)) and to estimate (4.7), it is possible to prove the following approximation result lim E |znµ (t) − zµ (t)|2H0 = 0,

n→∞

t ≥ 0.

An important consequence of this fact is that the semigroup P µ (t) can be approxiµ mated by the semigroup Pµn (t) associated with zn (t). Namely, for any ϕ ∈ Cb (H0 ) and t ≥ 0 it holds lim Pnµ (t)ϕ(z) = lim E ϕ(znµ (t)) = P µ (t)ϕ(z),

n→∞

n→∞

z ∈ H0 .

(5.10)

Now, for any u ∈ H we set   U (u) :=

O 0

u(x)

(5.11)

b(x, σ ) dσ dx.

Since b(x, ·) : R → R has linear growth, uniformly with respect to x ∈ O (see Hypothesis 2), it is not difficult to check that   |U (u)| ≤ c 1 + |u|2H , |U (u) − U (v)| ≤ c |u − v|H (1 + |u|H + |v|H ) , so that U : H → R is well defined and locally Lipschitz continuous. Moreover it is differentiable and DU (u) = b(·, u), for any u ∈ H . Hypothesis 3. The mapping U : H → R defined in (5.11) is bounded from above, that is sup U (u) < ∞. u∈ H

Remark 5.3. 1. The assumption of boundedness from above for U implies that  e 2U (u) N (0, (−
)−1 /2)(du) < ∞, Z := H

and

 Zn :=

e 2U (Pn u) N (0, (−
)−1 /2)(du) < ∞,

n ∈ N.

H

2. Hypothesis 3 above is satisfied if for example b(x, σ ) = −c1 (x) σ + c2 (x),

(x, σ ) ∈ O¯ × R,

for some continuous mappings c1 , c2 : O¯ → R, with minx∈ O¯ c1 (x) > 0.

On the Smoluchowski-Kramers approximation for a system

391

3. From the proof of Theorem 5.4 one sees that it is sufficient to assume a weaker condition than Hypothesis 3. Namely, what is needed is that both Z and Zn are finite and   lim ϕn (z)e 2U (Pn u) N (0, Cµ ) dz = ϕ(z)e 2U (u) N (0, Cµ ) dz, n→∞ H 0

H0

for any sequence {ϕn } ⊂ Cb (H0 ) uniformly bounded and pointwise convergent to some ϕ ∈ Cb (H0 ). Theorem 5.4. Assume that Hypotheses 1, 2 and 3 hold and take Q = I . Then the probability measure νµ (dz) :=

1 2U (u) e N (0, Cµ )(dz) Z

is invariant for system (5.1). Moreover the distribution 1 2U (u) N (0, (−
)−1 /2)(du) e Z is stationary for equation (4.1), for any µ > 0, and coincides with the unique invariant measure for the stochastic semi-linear heat equation (4.2). ν(du) :=

µ

Proof. For any µ > 0, n ∈ N and (q, p) ∈ Rn × Rn , we denote by ζn (t) := µ µ (qn (t), pn (t)) the solution of the system in Rn  µ µ µ  q˙n (t) = pn (t), qn (0) = q (5.12)  µ p˙ nµ (t) = Rn
Tn qnµ (t) + Rn b(·, Tn qnµ (t)) − pnµ (t) + W˙ n (t), pnµ (0) = p, where Wn (t) = (β1 (t), . . . , βn (t)), for any t ≥ 0. The transition semigroup associated with system (5.12) is defined for any ϕ ∈ Cb (R2n ) by

 Pˆnµ (t)ϕ(q, p) = E ϕ ζnµ (t) , t ≥ 0. Note that if we define Un (q) := U (Tn q), we have DUn (q) = Rn b(·, Tn q),

q ∈ Rn ,

and 1 D Rn
Tn q, qRn = Rn
Tn q 2 Moreover, since

q ∈ Rn .



 exp Rn
Tn q, qRn + 2 U (Tn q) dq Rn  = cn e2 U (Tn q) N (0, Rn (−
)−1 Tn /2) dq, Rn

for the obvious normalizing constant cn , by a change of variable from Hypothesis 3 we have

392

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S. Cerrai, M. Freidlin

 exp Rn
Tn q, qRn + 2 U (Tn q) dq = cn



< ∞.

e2 U (Pn u) N (0, (−
)−1 /2) du

H

As a well-known fact, the Boltzmann distribution  

 νˆ µ,n (dq, dp) = cµ,n exp Rn
Tn q, qRn + 2 Un (q) exp −µ|p|2Rn (dq, dp) =

1 2 Un (q) e N (0, Rn (−
)−1 Tn /2)(dq) × N (0, IRn /2µ)(dp) Zn

is invariant for system (5.12), so that for any ϕˆ ∈ Cb (Rn ) and t ≥ 0   Pˆnµ (t)ϕ(q, ˆ p) νˆ µ,n (dq, dp) = ϕ(q, ˆ p) νˆ µ,n (dq, dp). (5.13) Rn ×R n

Rn ×R n

µ Now, it is immediate to check that the H0 -valued process T¯n ζn (t) coincides with µ the solution zn (t) of the approximating system (5.9) with initial datum T¯n (q, p). For any ϕ ∈ Cb (H0 ) this yields

Pnµ (t)ϕ(T¯n (q, p)) = Pˆnµ (t)(ϕ ◦ T¯n )(q, p),

(q, p) ∈ Rn × Rn ,

and hence from (5.13) for any ϕ ∈ Cb (H0 ) we obtain  Pnµ (t)ϕ(T¯n (q, p)) νˆ µ,n (dq, dp) Rn ×Rn  ϕ(T¯n (q, p)) νˆ µ,n (dq, dp). = Rn ×Rn

(5.14)

If Tn is considered as a mapping from Rn into H by reasoning as above we have ( ) e2 Un (q) N (0, Rn (−
)−1 Tn /2) ◦ Tn−1 (du) = e2 U (u) N (0, (−
)−1 Pn /2)(du).

(5.15)

Moreover, if Tn is considered as a mapping from Rn into H −1 (O) we have N (0, IRn /2µ) ◦ Tn−1 = N (0, (−
)−1 Pn /2µ). Actually, for any λ ∈ H −1 (O) we have  )

( exp i λ, vH −1 (O) N (0, IRn /2µ) ◦ Tn−1 dv H −1 (O )      = exp i (−
)−1 λ, Tn p N (0, IRn /2µ) dp H Rn
  1  −1 −1 = exp − Rn (−
) λ, Rn (−
) λ n R 4µ
  1  = exp − (−
)−1 Pn λ, Pn λ −1 H (O ) 4µ 

 = exp i λ, vH −1 (O) N (0, (−
)−1 Pn /2µ) dv, H −1 (O )

and by uniqueness of the Fourier transform we obtain (5.16).

(5.16)

On the Smoluchowski-Kramers approximation for a system

393

Therefore, from (5.15) and (5.16) we have 1 2 U (u) νˆ µ,n ◦ T¯n−1 (dz) = e N (0, (−
)−1 Pn /2) × N (0, (−
)−1 Pn /2µ) (dz) Zn ) 1 2 U (u) ( N (0, Cµ ) ◦ P¯n−1 (dz), = e Zn and hence, since Pnµ (t)ϕ(z) = Pnµ (t)ϕ(P¯n z),

z ∈ H0 ,

from (5.14) it follows 1 Zn



Pnµ (t)ϕ(z) e2 U (Pn u) N (0, Cµ ) dz H0  1 = ϕ(P¯n z) e2 U (Pn u) N (0, Cµ ) dz. Zn H 0

(5.17)

Now, due to (5.8) and (5.10) we have lim Pnµ (t)ϕ(z) e2 U (Pn u) = P µ (t)ϕ(z) e2 U (u) .

n→∞

Then, thanks to Hypothesis 3, by the dominated convergence theorem we can take the limit as n goes to infinity in both sides of (5.17) and we get 1 Z

 µ

H0

P (t)ϕ(z) e

2 U (u)

1 N (0, Cµ ) dz = Z

 H0

ϕ(z) e2 U (u) N (0, Cµ ) dz,

for any ϕ ∈ Cb (H0 ). By a monotone class argument the same identity follows for arbitrary ϕ ∈ Bb (H0 ). This in particular implies that the measure νµ =

1 2 U (u) N (0, Cµ )(dz) e Z

is invariant for P µ (t). Finally, we obtain the second part of the theorem, as we have 1 [

1 2 U (u) 1 e N (0, Cµ )(dz)] = e2 U (u) N (0, (−
)−1 /2)(dz). Z Z  

Acknowledgements. Sandra Cerrai was supported by the PRIN Equazioni di Kolmogorov and Mark Freidlin by a NSF grant.

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